Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $t \neq 0$. $y = \dfrac{3t - 6}{-3t + 27} \div \dfrac{4t^2 - 4t - 8}{t^2 - 7t - 8} $
Answer: Dividing by an expression is the same as multiplying by its inverse. $y = \dfrac{3t - 6}{-3t + 27} \times \dfrac{t^2 - 7t - 8}{4t^2 - 4t - 8} $ First factor out any common factors. $y = \dfrac{3(t - 2)}{-3(t - 9)} \times \dfrac{t^2 - 7t - 8}{4(t^2 - t - 2)} $ Then factor the quadratic expressions. $y = \dfrac {3(t - 2)} {-3(t - 9)} \times \dfrac {(t + 1)(t - 8)} {4(t + 1)(t - 2)} $ Then multiply the two numerators and multiply the two denominators. $y = \dfrac {3(t - 2) \times (t + 1)(t - 8) } {-3(t - 9) \times 4(t + 1)(t - 2) } $ $y = \dfrac {3(t + 1)(t - 8)(t - 2)} {-12(t + 1)(t - 2)(t - 9)} $ Notice that $(t + 1)$ and $(t - 2)$ appear in both the numerator and denominator so we can cancel them. $y = \dfrac {3\cancel{(t + 1)}(t - 8)(t - 2)} {-12\cancel{(t + 1)}(t - 2)(t - 9)} $ We are dividing by $t + 1$ , so $t + 1 \neq 0$ Therefore, $t \neq -1$ $y = \dfrac {3\cancel{(t + 1)}(t - 8)\cancel{(t - 2)}} {-12\cancel{(t + 1)}\cancel{(t - 2)}(t - 9)} $ We are dividing by $t - 2$ , so $t - 2 \neq 0$ Therefore, $t \neq 2$ $y = \dfrac {3(t - 8)} {-12(t - 9)} $ $ y = \dfrac{-(t - 8)}{4(t - 9)}; t \neq -1; t \neq 2 $